田永革,郭文星.三个矩阵乘积厄米特Moore-Penrose逆的等式与不等式[J].数学研究及应用,2015,35(3):321~329
三个矩阵乘积厄米特Moore-Penrose逆的等式与不等式
Some Equalities and Inequalities for the Hermitian Moore-Penrose Inverse of Triple Matrix Product with Applications
投稿时间:2014-08-14  最后修改时间:2014-12-22
DOI:10.3770/j.issn:2095-2651.2015.03.010
中文关键词:  Moore-Penrose逆  反序律    惯量  偏序
英文关键词:Moore-Penrose inverse  reverse-order law  rank  inertia  L\"owner partial ordering
基金项目:国家自然科学基金 (Grant No.11271384).
作者单位
田永革 中央财经大学中国经济与管理研究院, 北京 100081 
郭文星 中央财经大学数学与统计学院, 北京 100081 
摘要点击次数: 979
全文下载次数: 1250
中文摘要:
      通过矩阵的秩和惯量公式研究矩阵广义逆之间的关系,建立了三个矩阵乘积的厄米特Moore-Penrose逆的一些等式与不等式.作为应用,给出了若干两个矩阵之和厄米特Moore-Penrose逆的等式与不等式.
英文摘要:
      We investigate relationships between the Moore-Penrose inverse $(ABA^{*})^{\dag}$ and the product $[(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$ through some rank and inertia formulas for the difference of $(ABA^{*})^{\dag} - [(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$, where $B$ is Hermitian matrix and $(AB)^{(1,2,3)}$ is a $\{1,\, 2,\,3\}$-inverse of $AB$. We show that there always exists an $(AB)^{(1,2,3)}$ such that $(ABA^{*})^{\dag}=[(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$ holds. In addition, we also establish necessary and sufficient conditions for the two inequalities $(ABA^{*})^{\dag} \succ [(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$ and $(ABA^{*})^{\dag} \prec [(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$ to hold in the L\"owner partial ordering. Some variations of the equalities and inequalities are also presented. In particular, some equalities and inequalities for the Moore-Penrose inverse of the sum $A + B$ of two Hermitian matrices $A$ and $B$ are established.
查看全文  查看/发表评论  下载PDF阅读器